Stability Analysis of Fractional Differential Systems with Order Lying in 1, 2

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doi:10.1155/2011/213485

Research Article

Stability Analysis of Fractional Differential Systems with Order Lying in 1, 2

Fengrong Zhang

^{1, 2}

and Changpin Li

¹

1Department of Mathematics, Shanghai University, Shanghai 200444, China

2School of Mathematics and Computational Science, China University of Petroleum (East China), Dongying 257061, China

Correspondence should be addressed to Changpin Li,[email protected] Received 6 December 2010; Revised 31 December 2010; Accepted 7 March 2011 Academic Editor: Dumitru Baleanu

Copyrightq2011 F. Zhang and C. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The stability ofn-dimensional linear fractional diﬀerential systems with commensurate order 1<

α <2 and the corresponding perturbed systems is investigated. By using the Laplace transform, the

asymptotic expansion of the Mittag-Leﬄer function, and the Gronwall inequality, some conditions on stability and asymptotic stability are given.

1. Introduction

Fractional calculus has a long history with more than three hundred years 1–3. Up to now, it has been proved that fractional calculus is very useful. Many mathematical models of real problems arising in various fields of science and engineering were established with the help of fractional calculus, such as viscoelastic systems, dielectric polarization, electrode- electrolyte polarization, and electromagnetic waves4–7.

Recently, the stability theory of fractional differential equations FDEs is of main interest in physical systems. Moreover, some stability results have been found8–17. These stability results are almost about the linear fractional differential systems with commensurate orderi.e., the fractional derivative order has to be an integer multiple of minimal fractional order18. For example, a necessary and sufficient condition on asymptotic stability of linear fractional differential system with order 0< α≤1 was first given in9. Then, some literatures on the stability of linear fractional differential systems with order 0 < α < 1 have been appeared11–15. However, not all the fractional differential systems have fractional orders in0, 1. There exist fractional models which have fractional orders lying in1, 2, for example, super-diffusion19. Hence, the stability of linear fractional differential systems with order

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1 < α < 2 has also been considered by using the conversion methods and transfer function 8,10. Almost all of the above literatures dealt with the fractional diﬀerential systems with Caputo derivative. Recently, Qian et al. 16 have investigated the stability of fractional diﬀerential systems with Riemann-Liouville derivative whose orderαlies in0, 1in details.

It is worth mentioning that not all of the stability conditions are parallel to the corresponding classical integer-order differential equations because of nonlocality and weak singularities of fractional calculus. For example, the solution to an autonomous fractional differential equation cannot define a dynamical system in the sense of semigroup20. Of course, some of the mathematical tools for the integer-order differential equation can be applied to fractional kinetics. In20, the authors first define the Lyapunov exponents for fractional differential system then determine their bounds, where the basic ideas and techniques are borrowed from21,22.

In this paper, we study the stability of autonomous linear fractional differential systems, nonautonomous linear fractional differential systems, and the corresponding perturbed systems with order 1 < α < 2 by using the properties of Mittag-Leffler functions and the Gronwall inequality.

The paper is organized as follows. InSection 2, we first recall some definitions and lemmas used throughout the paper. In Section 3, the stability analysis is presented for autonomous linear fractional diﬀerential systems with order 1 < α < 2. The stability of nonautonomous linear fractional diﬀerential systems and the corresponding perturbed systems are studied in Sections4and5, respectively. Conclusions and comments are included inSection 6.

2. Preliminaries

Let us denote by the set of real numbers, denote by the set of positive real numbers, denote by the set of positive integer numbers, and denote by the set of complex numbers.

In this section, we recall the most commonly used definitions and properties of fractional derivatives, Mittag-Leﬄer functions, and their asymptotic expansions.

Definition 2.1. The Riemann-Liouville derivative with orderαof functionxtis defined as follows:

RLD_t^α₀_,txt 1 Γm−α

d^m dt^m

_t

t0

t−τ^m−α−1xτdτ, 2.1

wherem−1≤α < m∈ ,Γ·is the Gamma function.

Definition 2.2. The Caputo derivative with orderαof functionxtis defined as follows:

CD^α_t₀_,txt 1 Γm−α

_t

t0

t−τ^m−α−1x^mτdτ, 2.2

wherem−1< α < m∈ .

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Their Laplace transforms fort00 are given as follows23:

L

RLD_0,t^α xt;s

s^αL{xt} −^m−1

k0

s^k

RLD^α−k−1_0,t xt

t0 m−1≤α < m, 2.3

L

CD^α_0,txt;s

s^αL{xt} −^m−1

k0

s^α−k−1x^k0 m−1< α≤m. 2.4

Definition 2.3. The Mittag-Leﬄer function is defined by

Eαz ^∞

k0

z^k

Γkα 1, 2.5

where the real part ofα, that is, α > 0,z∈. The two-parameter Mittag-Leﬄer function is defined by

Eα,βz ^∞

k0

z^k Γ

kα β , 2.6

where α >0 andβ∈,z∈.

One can seeEαz Eα,1zfrom the above equations. By analogy with2.6, forA∈

n×n, we introduce a matrix Mittag-Leﬄer function defined by24

Eα,βA ^∞

k0

A^k Γ

kα β . 2.7

The following definitions of stability are introduced.

Definition 2.4. The constantxeqis an equilibrium of fractional diﬀerential system^α_t₀_,txt ft, xif and only ifft, xeq α

t0,txt|_xtx_eq for allt > t0, where the operatort0,tdenotes either _RLD_t^α₀_,tor _CD_t^α₀_,t.

Without loss of generality, let the equilibrium bexeq 0, we introduce the following definition.

Definition 2.5. The zero solution of^α_t

0,txt ft, xtwith order 1 < α < 2 is said to be stable if, for any initial valuesxkk0,1, there existsε >0 such thatxt ≤εfor allt > t0. The zero solution is said to be asymptotically stable if, in addition to being stable,xt → 0 ast → ∞.

It is useful to recall the following asymptotic formulas for our developments in the sequel.

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Lemma 2.6. If 0< α < 2,βis an arbitrary complex number andμis an arbitrary real number such that

πα

2 < μ <min{π, πα}, 2.8

then for an arbitrary integerp≥1, the following expansions hold:

Eα,βz 1

αz^1−β/αexp z^1/α

−^p

k1

z^−k Γ

β−αk O

|z|^−p−1

, 2.9

with|z| → ∞,|argz| ≤μand

Eα,βz − p k1

z^−k Γ

β−αk O

|z|^−p−1

, 2.10

with|z| → ∞,μ≤ |argz| ≤π.

Proof. These results were proved in23.

Especially, taking into account the Lemma 2.6 and derivatives of the Mittag-Leﬄer function, we obtain

t^{αj β−1}E^j_α,βλt^α∼ ∂

∂λ _j

1

αλ^1−β/αexp

λ^1/αt , 2.11

witht → ∞,|argλ| ≤μand

t^{αj β−1}E^j_α,βλt^α∼−1^{j 1}

j!λ^−j−1 Γ

β−α t^β−α−1

j 1 !λ^−j−2 Γ

β−2α t^β−2α−1

, 2.12

witht → ∞,μ≤ |argz| ≤π,j0,1,2, . . . .

Lemma 2.7 see25. IfA ∈ ^n×n and 0 < α < 2, β is an arbitrary real number,μ satisfies πα/2< μ <min{π, πα}, andC >0 is a real constant, then

Eα,βA ≤ C

1 A, 2.13

whereμ≤ |argspecA| ≤ π, specAdenotes the eigenvalues of matrixAand · denotes the l2-norm.

Lemma 2.8Jordan Decomposition26. LetAbe a square complex matrix, then there exists an invertible matrixPsuch that

P⁻¹APJ1⊕ · · · ⊕Js, 2.14

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where theJlare the Jordan blocks ofAwith the eigenvalues ofAon the diagonal. The Jordan blocks are uniquely determined byA.

Lemma 2.9see27. If

xt≤ht _t

t0

ksxsds, t∈t0, T, 2.15

where all the functions involved are continuous ont0, T,T≤ ∞, andkt≥0, thenxtsatisfies

xt≤ht _t

t0

kshsexp _t

s

kudu

ds, t∈t0, T. 2.16

If, in addition,htis nondecreasing, then

xt≤htexp _t

t0

ksds

, t∈t0, T. 2.17

3. Stability of Autonomous Linear Fractional Differential Systems

3.1. The Riemann-Liouville Derivative Case

In this subsection, we consider the following system of fractional diﬀerential equations:

RLD^α_t₀_,txt Axt, t > t0, 3.1

with the initial conditions

RLD^α−k_t₀_,t xt|_tt₀xk−1 k1,2, 3.2

wherex∈ ⁿ, matrixA∈ ^n×n, and 1< α <2. Then, by analyzing the solutions of the above initial value problem3.1-3.2, one can find the following result.

Theorem 3.1. The autonomous fractional differential system3.1with Riemann-Liouville derivative and the initial conditions3.2is asymptotically stable iff|argspecA|> απ/2. In this case, the components of the state decay towards 0 liket^−α−1. Moreover, the system3.1is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy|argspecA|απ/2 have the same algebraic and geometric multiplicities.

Proof. Applying the Laplace transform, we can get the solution of3.1-3.2, xt t−t0^α−1Eα,α

At−t0^α x0 t−t0^α−2Eα,α−1

At−t0^α x1

1 k0

t−t0^α−k−1Eα,α−k

At−t0^α xk.

3.3

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Firstly, we study the properties of the elements of matrixest−t0^α−k−1·Eα,α−kAt−t0^α, k0,1. With regard to matrixA, there exists an invertible matrixP, such that

AP JP⁻¹PdiagJ1, J2, . . . , JsP⁻¹, 3.4

fromLemma 2.8, where the Jordan block

Jl

⎛

⎜⎜

⎜⎝ λl 1

λl 1 . .. ...

λl 1 λl

⎞

⎟⎟

⎟⎠

nl×nl

, 3.5

l 1,2, . . . , s,λl ∈ is the eigenvalue of matrixAand_s

l1nl n. Substituting 3.4into t−t0^α−k−1Eα,α−kAt−t0^α, we yield

At−t0^α t−t0^α−k−1P

∞ m0

diag

J₁^m, J₂^m, . . . , J_s^m t−t0^αm Γαm α−k P⁻¹

t−t0^α−k−1P

⎛

⎜⎜

⎜⎝ Eα,α−k

J1t−t0^α . ..

Eα,α−k

Jst−t0^α

⎞

⎟⎟

⎟⎠P⁻¹,

3.6

wherek0,1. The matrixt−t0^α−k−1Eα,α−kJlt−t0^αcan be written as follows by computing T

Eα,α−k

λt−t0^α

|_λλ_l, 3.7

where the operatorTis given as follows:

T t−t0^α−k−1

⎛

⎜⎜

⎝ 1 ∂

∂λ 1 2!

∂

∂λ 2

· · · 1 nl−1!

∂

∂λ nl−1

1 ∂

∂λ · · · 1 nl−2!

∂

∂λ _n_l₋₂

. .. . .. ...

1 ∂

∂λ 1

⎞

⎟⎟

⎠

. 3.8

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The nonzero elements oft−t0^α−k−1Eα,α−kJlt−t0^αcan be described uniformly as

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl

j1,2, . . . , nl . 3.9

iIfλl0, then

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl0

t−t0^jα−k−1 Γ

jα−k . 3.10

It is obvious thatt−t0^jα−k−1/Γjα−k → ∞t → ∞fork0 andj ≥1. Thus,xt →

∞t → ∞.

iiIfλl/0, three cases will be considered separately.

Case 1|argspecA||argλl|> απ/2. If|argλl|> απ/2 andt → ∞, then

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl

∼−1^j

⎡

⎣λ^−j_l t−t0^−k−1 Γ−k

jλ^−j−1_l t−t0^{−k−α−1} Γ−k−α

⎤

⎦.

3.11

That is to say,t−t0^α−k−11/j−1!{∂/∂λ^j−1Eα,α−kλt−t0^α}|λλl → 0t → ∞from the asymptotic expansion2.12andxt → 0t → ∞. Moreover, the components of the state decay towards 0 liket^−α−1. Taking into account the entire functionE_α,α−kλt−t0^α, we also get the boundedness oft−t0^α−k−1/j −1!{∂/∂λ^j−1Eα,α−kλt−t0^α}|_λλ_l j 1,2, . . . , nl;l1,2, . . . , s.

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl

t−t0^jα−k−1

j−1 ! E_α,α−k^j−1

λt−t0^α

λλl

∼ 1 j−1 !

∂

∂λ _j−1

1

αλ^{1−α k/α}e^λ^1/α^t−t⁰

λλl

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1 j−1 !

1 k−α1 k−2α· · ·

1 k− j−1 α

α^j λ^{1 k−jα/α}_l · · ·

j

j−1 /2−

j−1 α−k−

j−1 j−2 /2α α^j

×λ^{j1−α k−1/α}_l t−t0^j−2 1

α^jλ^{j1−α k/α}_l t−t0^j−1

"

exp

λ^1/α_l t−t0 , 3.12

then

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

E_α,α−k

λt−t0^α

λλl

∼ 1 j−1 !

1 k−α1 k−2α· · ·

1 k− j−1 α

α^j λ^{1 k−jα/α}_l · · · j

j−1 /2−

j−1 α−k−

j−1 j−2 /2 α

α^j λ^{j1−α k−1/α}_l t−t0^j−2 1

α^jλ^{j1−α k/α}_l t−t0^j−1 exp

#

|λl|^1/αcos

argλl α

t−t0

"

−→ ∞ ast−→ ∞, j1,2, . . . , nl,

3.13

because of|argλl/α|< π/2, that is, cosargλ l/α>0.

So,xt₁

k0t−t0^α−k−1Eα,α−kAt−t0^αxk → ∞t → ∞.

Case 3|argspecA||argλl|απ/2. Letλl rcosαπ/2 isinαπ/2, whereris the modulus ofλlandi²−1.

Firstly, suppose that the critical eigenvalueλl has the same algebraic and geometric multiplicities, that is, the matrixJlis a diagonal matrix, then, according to3.7, we have

Jlt−t0^α t−t0^α−k−1Eα,α−k

λlt−t0^α diag1, . . . ,1. 3.14

If|argλl|απ/2, we have the diagonal elements of matrix3.14|t−t0^α−k−1Eα,α−kλlt− t0^α|∼ 1/αr^{1 k−α/α} t → ∞from the asymptotic expansion2.11. So, the solution of 3.1is stable in this case.

Next, suppose that the algebraic multiplicity of critical eigenvalue λl is not equal to the geometric multiplicity, that is, the matrix Jl is a Jordan block matrix, and matrix

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t− t0^α−k−1Eα,α−kJlt− t0^α is the same as 3.7, then the nondiagonal elements of t − t0^α−k−1Eα,α−kJlt−t0^αcan be evaluated from3.12as follows:

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl

∼ 1 j−1 !

1 k−α1 k−2α· · ·

1 k− j−1 α

α^j λ^{1 k−jα/α}_l · · · j

j−1

2 −

j−1 α−k−

j−1 j−2 /2 α

α^j λ^{j1−α k−1/α}_l t−t0^j−2 1

"

exp

ir^1/αt−t0

, j2, . . . , nl.

3.15

So,

t−t0^α−k−1 1 j−1 !

∂

∂λ _j−1

Eα,α−k

λt−t0^α

λλl

∼ 1 j−1 !

1 k−α1 k−2α· · ·

1 k− j−1 α

α^j λ^{1 k−jα/α}_l · · · j

j−1 /2−

j−1 α−k−

j−1 j−2 /2 α

α^j λ^{j1−α k−1/α}_l t−t0^j−2 1

−→ ∞ ast−→ ∞, j2, . . . , nl,

3.16

that is,xt → ∞ ast → ∞.

According to the above discussions, the proof is completed.

Remark 3.2. 1If|argspecA|< απ/2, then system3.1is not stable.

2IfAhas zero eigenvalue, system3.1is not stable.

3IfAhas critical eigenvaluesλc, that is,|argλc| απ/2, and the arguments of the rest eigenvalues in absolute values are greater thanαπ/2, then system3.1is not stable provided thatλchas diﬀerent geometric and algebraic multiplicities.

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3.2. The Caputo Derivative Case

Now, we consider the fractional diﬀerential system with Caputo derivative

CD^α_t₀_,txt Axt, t > t0, 3.17

under the initial conditions

x^kt0 xk k0,1, 3.18

wherex,A, andαare as inSection 3.1. Then, one can get the following theorem.

Theorem 3.3. The autonomous fractional differential system 3.17 with Caputo derivative and initial conditions 3.18 is asymptotically stable iff |argspecA| > απ/2. In this case, the components of the state decay towards 0 liket^{−α 1}. Moreover, the system3.17 is stable iff either it is asymptotically stable, or those critical eigenvalues which satisfy|argspecA| απ/2 have the same algebraic and geometric multiplicities.

Proof. This theorem can be proved in the same manner as that in the proof ofTheorem 3.1, so it is omitted here.

4. Stability of Nonautonomous Linear Fractional Differential Systems

4.1. The Riemann-Liouville Derivative Case

We will consider a nonautonomous fractional diﬀerential system with Riemann-Liouville derivative

RLD^α_t₀_,txt Axt Btxt, t > t0, 4.1

RLD^α−k_t₀_,t xt

tt0

xk−1 k1,2, 4.2

wherex∈ ⁿ, matrixA∈ ^n×n,Bt:t0,∞ → ^n×n is a continuous matrix, and 1< α < 2.

The main results of this subsection are derived as follows.

t0 Btdtis bounded, then the zero solution of 4.1is stable.

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Proof. Applying the Laplace transform, we can get the solution of4.1-4.2,

xt t−t0^α−1Eα,α

At−t0^α x1

_t

t0

t−τ^α−1Eα,α

At−τ^α Bτxτdτ

1 k0

At−t0^α xk

_t

t0

At−τ^α Bτxτdτ.

4.3

From the proof ofTheorem 3.1, the matrixt−t0^α−k−1Eα,α−kAt−t0^αis bounded fork0,1.

Therefore, there exist positive numbersMk, such thatt−t0^α−k−1Eα,α−kAt−t0^α ≤Mk

k0,1. Now, we can get the estimate of solutionxt

xt ≤M0x0 M1x1 _t

t0

M0Bτ · xτdτ. 4.4

Applying the Gronwall inequality2.17leads to

xt ≤M0x0 M1x1exp

M0

_t

t0

Bτdτ

. 4.5

Thus, we derive thatxtis bounded according to the condition$_∞

t0 Btdt < ∞, that is, the zero solution of4.1is stable. The proof is completed.

Similarly, we can derive the following conclusion.

Theorem 4.2. If the matrix A such that|specA|/0, |argspecA| > απ/2, andBt Ot−t0^γ(−1< γ <1−α,t0>0) fort≥t0, then the zero solution of4.1is asymptotically stable.

Proof. From the proof ofTheorem 3.1, the following expression is valid:

xt ≤t−t0^α−2L1x0 t−t0^α−2L2x1 L1

_t

t0

t−τ^α−2Bτ · xτdτ, 4.6

whereL1, L2>0 such thatt−t0Eα,αAt−t0^α< L1andE_α,α−1At−t0^α< L2. Moreover, from4.4and2.17, one has

xt ≤M0x0 M1x1 L1

_t

t0

t−τ^α−2Bτ · xτdτ

≤M0x0 M1x1exp

L1

_t

t0

t−τ^α−2Bτdτ

.

4.7

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Substituting4.7into4.6, we have

xt ≤t−t0^α−2L1x0 L2x1 M01

_t

t0

t−τ^α−2Bτe^L¹^$^t^τ⁰^τ−η^α−2^Bηdηdτ, 4.8

whereM01L1M0x0 M1x1. It follows from the conditionBtOt−t0^γ−1< γ <

1−α,t0>0fort≥t0that there exists a constantM >0, such that$_t

t0t−τ^α−2Bτdτ < M and

xt ≤t−t0^α−2L1x0 L2x1 M01e^L¹^M _t

t0

t−τ^α−2Oτ−t0^γdτ t−t0^α−2L1x0 L2x1 M01e^L¹^MΓα−1Γ

1 γ Γ

α γ Ot−t0^{γ α−1}.

4.9

So, the zero solution of4.1is asymptotically stable.

4.2. The Caputo Derivative Case

In this subsection, we consider a nonautonomous fractional diﬀerential system with Caputo derivative

CD^α_t₀_,txt Axt Btxt, t > t0, 4.10

x^kt0 xk k0,1, 4.11

wherex,A, andαare as inSection 4.1,Bt:t0,∞ → ^n×n is a continuously diﬀerentiable matrix. We can get the solution of4.10-4.11by using the Laplace transform and Laplace inverse transform

xt Eα

At−t0^α x0 t−t0Eα,2

At−t0^α x1

_t

t0

At−τ^α Bτxτdτ. 4.12

The main stability results of this subsection are derived as follows.

t0 Btdtis bounded, then the zero solution of 4.10is stable.

Proof. The proof line is similar to that ofTheorem 4.1.

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Theorem 4.4. If the matrix A such that|specA|/0, |argspecA| > απ/2, andBt Ot−t0^γ (−1< γ <1−α,t0>0) fort≥t0, then the zero solution of 4.10is asymptotically stable.

Proof. From the solution4.12andLemma 2.7, we can directly get

xt ≤ C0x0 1 At−t0^α

C1t−t0x1 1 At−t0^α L1

_t

t0

t−τ^α−2Bτ · xτdτ, 4.13

whereC0, C1>0 andL1>0, such thatt−t0Eα,αAt−t0^α< L1. Furthermore, there exists a constantM0 >0 such that

C0x0 1 At−t0^α

C1t−t0x1

1 At−t0^α ≤M0, 4.14

that is,

xt ≤M0 L1

_t

t0

t−τ^α−2Bτ · xτdτ

≤M0exp

L1

_t

t0

t−τ^α−2Bτdτ

.

4.15

Substituting4.15into4.13gives

C1t−t0x1 1 At−t0^α L1M0

_t

t0

t−τ^α−2Bτe^L¹^$^t^τ⁰^τ−η^α−2^Bηdηdτ.

4.16

It follows from the conditionBtOt−t0^γ −1< γ < 1−α,t0 >0fort ≥t0that there exists a constantM >0, such that$_t

t0t−τ^α−2Bτdτ < Mand

C1t−t0x1

1 At−t0^α L1M0e^L¹^M _t

t0

t−τ^α−2Oτ−t0^γdτ

C0x0 1 At−t0^α

C1t−t0x1

1 At−t0^α L1M0e^L¹^MΓα−1Γ 1 γ Γ

α γ Ot−t0^{γ α−1}. 4.17 So, the zero solution of4.10is asymptotically stable.

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5. Stability of the Perturbed Systems

In this section, we only study the perturbed system of a linear fractional diﬀerential system with Riemann-Liouville derivative

RLD^α_t₀_,txt Axt ft, xt, t > t0, 5.1 under the initial conditions

RLD^α−k_t₀_,t xt

tt0

xk−1 k1,2, 5.2

wherex∈ ⁿ, matrixA∈ ^n×n, and 1 < α < 2.ft, x :t0,∞× ⁿ → ⁿ is a continuous function in whichft,0 0; moreover,ft, xfulfils the Lipschitz condition with respect to x. Then, the unique solution of5.1-5.2can be written as

xt t−t0^α−1Eα,α

At−t0^α x1

_t

t0

At−τ^α fτ, xτdτ. 5.3

The following theorem can be proved by the same argument used in the proof of Theorem 4.1.

Moreover, suppose that there exists a positive functionγtwhich satisfies the following conditions:

i$_∞

t0 γtdtis bounded, iift, x ≤γtxt,

then the zero solution of 5.1is stable.

Theorem 5.2. If the matrixAsuch that|specA|/0,|argspecA| > απ/2,α 1/A < 2, and suppose that the functionft, xsatisfies uniformly

xlim→0

ft, x

x 0, t∈t0,∞, 5.4

then the zero solution of 5.1is asymptotically stable.

Proof. According to the proof ofTheorem 3.1andLemma 2.7, we have

xt ≤t−t0^α−2L1x0 t−t0^α−2L2x1 _t

t0

C1t−τ^α−1

1 At−τ^αfτ, xτdτ, 5.5

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whereC1, L1, L2 > 0 such thatt−t0Eα,αAt−t0^α < L1 andEα,α−1At−t0^α < L2. Taking into account the condition5.4, there exists a constantδ >0, such that

ft, xt< 1

C1xt asxt< δ. 5.6

Then,

xt ≤Mt−t0^α−2 _t

t0

t−τ^α−1

1 At−τ^αxτdτ, 5.7 whereML1x0 L2x1. Applying the Gronwall inequality2.16to5.7yields

xt ≤Mt−t0^α−2 M _t

t0

t−τ^α−1τ−t0^α−2

1 At−τ^α e^$^τ^t^t−s^α−1^{/1 At−s}^α^dsdτ Mt−t0^α−2 M

_t

t0

t−τ^α−1τ−t0^α−2

1 At−τ^α ^1−1/αAdτ

≤Mt−t0^α−2 MA^1/αA−1

_t

t0

t−τ^1/A−1τ−t0^α−2dτ

Mt−t0^α−2 MA^1/αA−1Γ1/AΓα−1

Γα 1/A −1 t−t0^{1/A α−2}.

5.8

So, the zero solution of5.1is asymptotically stable due toα 1/A −2 < 0. The proof is thus finished.

6. Conclusion

It is well know that many physical phenomena having memory and genetic characteristics can be described by using the fractional differential systems. Especially, the fractional differential systems with order 1 < α < 2 have recently gained an increasing attention 23,28–31. It should be noted that29,30 are earlier and interesting work on fractional interval systems. Motivated by the above research activities, in this paper, we have studied the stability of linear fractional differential systems and the corresponding perturbed systems with Rimann-Liouville derivative and Caputo derivative for the commensurate order 1 <

α < 2. The main analytic tools used in this paper are the Mittag-Leffler function and the Gronwall inequality. For the autonomous linear fractional differential systems with order 1 < α < 2, the necessary and sufficient conditions on stability and asymptotic stability are given, which are almost the same as those with the fractional derivative orderα ∈ 0, 1.

But the components of the state decay towards 0 liket^{−α 1}, which is different from the case with Caputo derivative orderα∈0,1. For the nonautonomous linear fractional differential systems, we have derived some sufficient conditions on stability and asymptotic stability.

We have further given the asymptotic stability results of the perturbed systems with order 1< α <2.

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Acknowledgments

The authors wish to thank Professors D. Baleanu and Juan J. Trujillo for their kind invitation to submit our paper to their special issue on Fractional Models and their Applications. They also thank the anonymous reviewers of this paper for their careful reading and invaluable correction suggestions. The present work was partially supported by the National Natural Science Foundation of China under Grant no. 10872119 and the the Key Disciplines of Shanghai Municipality under Grant no. S30104.

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